423
Preliminaries
The following notations will be used throughout the paper.
R
denotes the set of all non-negative real numbers,
Z
denotes the set of all non-negative integers,
n
R
denotes the n-finite-dimensional Euclidean space with the
Euclidean norm
.
, the scalar product between
x
and
y
is defined by
,
T
x y
n m
u
R
denotes the set of all
(
)
n m
u
-matrices, and
T
A
denotes the transpose of the matrix
A
, Matrix
n n
Q
u
R
is positive semidefinite
(
0)
Q
t
if
0,
T
x Qx
t
.
n
x
R
If
0(
0
T
T
x Qx x Qx
!
, resp.)
0
x
z
, then
Q
is positive
(negative, resp.) definite and denoted by
0, (
0,
Q Q
!
resp.). It is easy to verify that
0,
Q
!
(
0,
Q
resp.) iff
,
n
x
R
0,
E
!
2
T
x Qx x
E
t
(
,
0,
n
x
E
!
R
2
,
,
T
n
x Qx
x x
E
d
R
resp.).
Lemma 2.1
(Hale 1977) The zero solution of difference system is asymptotic stability if there exists a positive
definite function
( ) :
n
V x
o
R R
such that
2
0 :
( ( ))
( ( 1)) ( ( ))
( ) ,
V x k V x k
V x k
x k
E
E
! '
d
along the solution of the system. In case the above condition holds for all
( )
x k V
G
, we say that the zero
solution is locally asymptotically stable.
Lemma 2.2
For any constant symmetric matrix
n n
M
u
,
0
T
M M
!
, scalar
/{0}
s
, vector function
:[0, ]
n
W s
o
, we have
1
1
1
0
0
0
( ( ) ( ))
( )
( ) .
T
s
s
s
T
i
i
i
s w i Mw i
w i M w i
§
· §
·
t ¨
¸ ¨
¸
©
¹ ©
¹
¦
¦ ¦
We present the following technical lemmas, which will be used in the proof of our main result.
Main results
In this section, we consider the sufficient condition for asymptotic stability of the zero solution
v
of
(1) in terms of certain matrix inequalities. Without loss of generality, we can assume that
*
0, (0) 0
v
S
and
f
=0 (for otherwise, we let
*
x v v
and define
*
*
( ) (
) ( ))
S x S x v S v
.
